Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CONVOLUTIONS OF WHITE NOISE OPERATORS

  • Ji, Un-Cig (Department of Mathematics Research Institute of Mathematical Finance Chungbuk National University) ;
  • Kim, Young-Yi (Department of Mathematics Chungbuk National University)
  • Received : 2010.02.24
  • Published : 2011.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Motivated by the convolution product of white noise functionals, we introduce a new notion of convolution products of white noise operators. Then we study several interesting relations between the convolution products and the quantum generalized Fourier-Mehler transforms, and study a quantum-classical correspondence.

Keywords

References

  1. M. Ben Chrouda, M. El Ouled, and H. Ouerdiane, Quantum stochastic processes and applications, Quantum probability and infinite dimensional analysis, 115-125, QP-PQ: Quantum Probab. White Noise Anal., 18, World Sci. Publ., Hackensack, NJ, 2005.
  2. D. M. Chung and U. C. Ji, Transforms on white noise functionals with their applications to Cauchy problems, Nagoya Math. J. 147 (1997), 1-23. https://doi.org/10.1017/S0027763000006292
  3. D. M. Chung and U. C. Ji, Transformation groups on white noise functionals and their applications, Appl. Math. Optim. 37 (1998), no. 2, 205-223. https://doi.org/10.1007/s002459900074
  4. D. M. Chung, U. C. Ji, and N. Obata, Quantum stochastic analysis via white noise operators in weighted Fock space, Rev. Math. Phys. 14 (2002), no. 3, 241-272. https://doi.org/10.1142/S0129055X0200117X
  5. T. Hida, Analysis of Brownian Functionals, Carleton Math. Lect. Notes no. 13, Carleton University, Ottawa, 1975.
  6. R. L. Hudson and K. R. Parthasarathy, Quantum Ito's formula and stochastic evolu- tions, Comm. Math. Phys. 93 (1984), no. 3, 301-323. https://doi.org/10.1007/BF01258530
  7. U. C. Ji, Quantum extensions of Fourier-Gauss and Fourier-Mehler transforms, J. Korean Math. Soc. 45 (2008), no. 6, 1785-1801. https://doi.org/10.4134/JKMS.2008.45.6.1785
  8. U. C. Ji and N. Obata, Quantum white noise calculus, in "Non-Commutativity, Infinite- Dimensionality and Probability at the Crossroads (N. Obata, T. Matsui and A. Hora, Eds.)," pp. 143-191, World Scientific, 2002.
  9. U. C. Ji and N. Obata, A unified characterization theorem in white noise theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 2, 167-178. https://doi.org/10.1142/S0219025703001122
  10. U. C. Ji and N. Obata, Annihilation-derivative, creation-derivative and representation of quantum martingales, Comm. Math. Phys. 286 (2009), no. 2, 751-775. https://doi.org/10.1007/s00220-008-0702-3
  11. H.-H. Kuo, White Noise Distribution Theory, CRC Press, 1996.
  12. P.-A. Meyer, Quantum Probability for Probabilists, Lect. Notes in Math. Vol. 1538, Springer-Verlag, 1993.
  13. N. Obata, White Noise Calculus and Fock Space, Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994.
  14. N. Obata, An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan 45 (1993), no. 3, 421-445. https://doi.org/10.2969/jmsj/04530421
  15. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhauser, 1992.

Cited by

  1. YEH CONVOLUTION OF WHITE NOISE FUNCTIONALS vol.31, pp.5_6, 2013, https://doi.org/10.14317/jami.2013.825
  2. STOCHASTIC DIFFERENTIAL EQUATION FOR WHITE NOISE FUNCTIONALS vol.29, pp.2, 2016, https://doi.org/10.14403/jcms.2016.29.2.337
  3. Factorization property of convolutions of white noise operators vol.46, pp.4, 2015, https://doi.org/10.1007/s13226-015-0146-3